collective additive tree spanners of homogeneously

Collective Additive Tree Spanners ofHomogeneously Orderable Graphs

[Extended Abstract]

Feodor F. Dragan, Chenyu Yan, and Yang Xiang

Algorithmic Research Laboratory, Department of Computer ScienceKent State University, Kent, OH 44242, USA

[email protected], [email protected], [email protected]

Abstract. In this paper we investigate the (collective) tree spannersproblem in homogeneously orderable graphs. This class of graphs was in-troduced by A. Brandstadt et al. to generalize the dually chordal graphsand the distance-hereditary graphs and to show that the Steiner treeproblem can still be solved in polynomial time on this more general classof graphs. In this paper, we demonstrate that every n-vertex homoge-neously orderable graph G admits– a spanning tree T such that, for any two vertices x, y of G, dT (x, y) ≤

dG(x, y) + 3 (i.e., an additive tree 3-spanner) and– a system T (G) of at most O(log n) spanning trees such that, for

any two vertices x, y of G, a spanning tree T ∈ T (G) exists withdT (x, y) ≤ dG(x, y) + 2 (i.e, a system of at most O(log n) collectiveadditive tree 2-spanners).

These results generalize known results on tree spanners of dually chordalgraphs and of distance-hereditary graphs. The results above are alsocomplemented with some lower bounds which say that on some n-vertexhomogeneously orderable graphs any system of collective additive tree1-spanners must have at least Ω(n) spanning trees and there is no systemof collective additive tree 2-spanners with constant number of trees.

1 Introduction

A spanning tree T of a graph G is called a tree spanner of G if T provides a“good” approximation of the distances in G. More formally, for r ≥ 0, T is calledan additive tree r-spanner of G if for any pair of vertices u and v their distancein T is at most r plus their distance in G (see [17,19]). A similar definition canbe given for multiplicative tree t-spanners (see [6]); however in this paper we areonly concerned with additive spanners. Tree spanners have many applications invarious areas. They occur in biology and they can be used in message routing andas models for broadcast operations in communication networks. Tree spannersare useful also from the algorithmic point of view - many algorithmic problemsare easily solvable on trees. If one needs to solve an NP -hard optimizationproblem concerning distances in a graph G and G admits a good tree spanner

E.S. Laber et al. (Eds.): LATIN 2008, LNCS 4957, pp. 555–567, 2008.c© Springer-Verlag Berlin Heidelberg 2008

556 F.F. Dragan, C. Yan, and Y. Xiang

T , then an efficient solution to that problem on T would provide an approximatesolution to the original problem on G.

The problem to decide for a graph G whether G has a multiplicative treet–spanner (the multiplicative tree t–spanner problem) is NP–complete for anyfixed t ≥ 4 [6], and it remains NP -complete even on some rather restricted graphfamilies. Fortunately, there is also a number of special classes of graphs whereadditive or multiplicative variants of the tree spanner problem are polynomialtime solvable. Here we will mention only the results on two families of graphswhich are relevant to our paper. Every dually chordal graph admits an additivetree 3-spanner [2] and every distance-hereditary graph admits an additive tree2-spanner [19], and such tree spanners can be constructed in linear time.

In [12] we generalized the notion of tree spanners by defining a new notion ofcollective tree spanners. We say that a graph G admits a system of μ collectiveadditive tree r-spanners if there is a system T (G) of at most μ spanning trees ofG such that for any two vertices u, v of G a spanning tree T ∈ T (G) exists suchthat the distance in T between u and v is at most r plus their distance in G. Wesay that system T (G) collectively r-spans the graph G and r is the (collective)additive stretch factor. Clearly, if G admits a system of μ collective additivetree r-spanners, then G admits an additive r-spanner with at most μ × (n − 1)edges, and if μ = 1 then G admits an additive tree r-spanner. Note that, aninduced cycle of length k provides an example of a graph which does not haveany additive tree (k − 3)-spanner, but admits a system of two collective additivetree 0-spanners. Furthermore, for any r ≥ 1 there is a chordal graph whichdoes not have any additive tree r-spanner [19]; on the other hand, any n-vertexchordal graph admits a system of O(log n) collective additive tree 2-spanners [12].These two examples demonstrate the power of this new concept of collective treespanners. One of the motivations to introduce this new concept stems from theproblem of designing compact and efficient routing schemes in graphs. In [14,20],a shortest path routing labeling scheme for trees is described that assigns eachvertex of an n-vertex tree a O(log2 n/ log log n)-bit label. Given the label of asource vertex and the label of a destination, it is possible to compute in constanttime, based solely on these two labels, the neighbor of the source that heads inthe direction of the destination. Clearly, if an n-vertex graph G admits a systemof μ collective additive tree r-spanners, then G admits a routing labeling schemeof deviation (i.e., additive stretch) r with addresses and routing tables of sizeO(μ log2 n/ log log n) bits per vertex. Once computed by the sender in μ time,headers of messages never change, and the routing decision is made in constanttime per vertex (for details see [11,12]). Other motivations stem from the genericproblems of efficient representation of the distances in ”complicated” graphsby the tree distances and of algorithmic use of these representations [1,7,13].Approximating graph distance dG by a simpler distance (in particular, by tree–distance dT ) is useful in many areas such as communication networks, dataanalysis, motion planning, image processing, network design, and phylogeneticanalysis.

Collective Additive Tree Spanners of Homogeneously Orderable Graphs 557

Previously, collective tree spanners of particular classes of graphs were con-sidered in [8,10,11,12,16]. Paper [12] showed that any chordal graph or chordalbipartite graph admits a system of at most log2 n collective additive tree 2–spanners. These results were complemented by lower bounds, which say thatany system of collective additive tree 1–spanners must have Ω(

√n) spanning

trees for some chordal graphs and Ω(n) spanning trees for some chordal bipar-tite graphs. Furthermore, it was shown that any c-chordal graph admits a systemof at most log2 n collective additive tree (2�c/2�)–spanners and any circular-arcgraph admits a system of two collective additive tree 2–spanners. Paper [11]showed that any AT-free graph (graph without asteroidal triples) admits a sys-tem of two collective additive tree 2-spanners and any graph having a dominatingshortest path admits a system of two collective additive tree 3-spanners and asystem of five collective additive tree 2-spanners. In paper [8], it was shown thatno system of constant number of collective additive tree 1-spanners can exist forunit interval graphs, no system of constant number of collective additive treer-spanners can exist for chordal graphs and r ≤ 3, and no system of constantnumber of collective additive tree r-spanners can exist for weakly chordal graphsand any constant r. On the other hand, [8] proved that any interval graph ofdiameter D admits an easily constructible system of 2 log(D − 1) + 4 collectiveadditive tree 1-spanners.

Only papers [10,16] have investigated (so far) collective tree spanners in theweighted graphs. Paper [10] demonstrated that any weighted graph with tree-width at most k−1 admits a system of k log2 n collective additive tree 0–spanners,any weighted graph with clique-width at most k admits a system of k log3/2 n col-lective additive tree (2w)–spanners, and any weighted graph with size of largestinduced cycle at most c (i.e., a c-chordal graph) admits a system of 4 log2 ncollective additive tree (2(�c/3� + 1)w)–spanners (here, w is the maximum edgeweight in G). The latter result was refined for weighted weakly chordal graphs:any such graph admits a system of 4 log2 n collective additive tree (2w)-spanners.In [16], it was shown that any n–vertex planar graph admits a system of O(

√n)

collective multiplicative tree 1-spanners (equivalently, additive tree 0-spanners)and a system of at most 2 log3/2 n collective multiplicative tree 3–spanners.

In this paper we study collective additive tree spanners in homogeneously or-derable graphs. The class of homogeneously orderable graphs was introduced in[4] to generalize the dually chordal graphs and the distance-hereditary graphsand to show that the Steiner tree problem can still be solved in polynomialtime on this more general class of graphs. The follow up to [4] paper [9] showedalso that both the connected r-domination problem and the r-dominating cliqueproblem are polynomial time solvable on homogeneously orderable graphs. Ho-mogeneously orderable graphs (HOGs) is a large family of graphs which com-prises a number of well-known graph classes including Dually Chordal graphs,House-Hole-Domino-Sun-free graphs (HHDS-free graphs), Distance-Hereditarygraphs, Strongly Chordal graphs, Interval graphs and others (see Figure 1). InSection 3, we show that every homogeneously orderable graph admits an ad-ditive tree 3-spanner constructible in linear time. In Section 4, we demonstrate


HOGs

dually chordal graphs

strongly chordal graphs

directed path graphs ptolemaic graphs

interval graphs trees

distance−hereditary graphs

HHDS−free graphs

doubly chordal graphs

Fig. 1. Hierarchy of Homogeneously Orderable Graphs. For definitions of graph classesincluded see [5].

that every homogeneously orderable graph admits a system of O(log n) collectiveadditive tree 2-spanners constructible in polynomial time. These results gener-alize known results on tree spanners of dually chordal graphs and of distance-hereditary graphs (see [2] and [19], respectively). Table 1 summarizes the resultsof this paper.

Table 1. Collective additive tree spanners of n-vertex homogeneously orderable graphs

additive stretch upper bound on lower bound onfactor number of trees number of trees

3 1 12 log2 n c < μ ≤ log2 n1 n − 1 Ω(n)0 n − 1 Ω(n)

2 Preliminaries

All graphs occurring in this paper are connected, finite, undirected, unweighted,loopless and without multiple edges. In a graph G = (V, E) (n = |V |, m = |E|)the length of a path from a vertex v to a vertex u is the number of edges inthe path. The distance dG(u, v) between the vertices u and v is the length of ashortest path connecting u and v. The i-th neighborhood of a vertex v of G is theset Ni(v) = {u ∈ V : dG(v, u) = i}. For a vertex v of G, the sets N(v) = N1(v)and N [v] = N(v) ∪ {v} are called the open neighborhood and the closed neigh-borhood of v, respectively. For a set S ⊆ V , by N [S] =

⋃v∈S N [v] we denote


the closed neighborhood of S and by N(S) = N [S] \ S the open neighborhood ofS. The disk of radius k centered at v is the set of all vertices of distance at mostk to v, i.e., Dk(v) = {u ∈ V : dG(u, v) ≤ k} =

⋃{Ni(v) : i = 0, . . . , k}.

Denote by D(G) = {Dr(v) : v ∈ V , r a non-negative integer} the family ofall possible disks of G and by L(D(G)) the intersection graph of those disks, i.e.,the vertices of L(D(G)) are disks from D(G) and two vertices are adjacent ifand only if the corresponding disks share a common vertex. Note that two disksDp(x) and Dq(y) intersect if and only if dG(x, y) ≤ p + q. The k–th power Gk ofa graph G = (V, E) is the graph with vertex set V and edges between verticesu, v with distance dG(u, v) ≤ k. In what follows, a subset U of V is a k–set if Uinduces a clique in the power Gk, i.e., for any pair x, y of vertices of U we havedG(x, y) ≤ k. A graph G is called chordal if it does not have any induced cycleof length greater than 3.

We say that a graph G = (V, E) admits a system of μ collective additive treer-spanners if there is a system T (G) of at most μ spanning trees of G suchthat for any two vertices x, y of G a spanning tree T ∈ T (G) exists such thatdT (x, y) ≤ dG(x, y)+ r. If μ = 1, then the tree T such that T (G) = {T } is calleda tree r-spanner of G.

A nonempty set H ⊆ V is homogeneous in G = (V, E) if all vertices of H havethe same neighborhood in V \H , i.e., N(u)∩(V \H) = N(v)∩(V \H) for all u, v ∈H, (any vertex w ∈ V \ H is adjacent to either all or none of the vertices fromH). A homogeneous set H is proper if |H | < |V |. Trivially for each v ∈ V thesingleton {v} is a proper homogeneous set. Let U1, U2 be disjoint subsets of V .If every vertex of U1 is adjacent to every vertex of U2 then U1 and U2 form ajoin, denoted by U1 �� U2. A set U ⊆ V is join–split if U can be partitioned intotwo nonempty sets U1, U2 such that U = U1 �� U2.

Next we recall the definition of homogeneously orderable graphs as given in[4]. A vertex v of G = (V, E) with |V | > 1 is h–extremal if there is a propersubset H ⊂ D2(v) which is homogeneous in G and for which D2(v) ⊆ N [H ]holds, i.e., H dominates D2(v). Thus, the sets H and D2(v) \ H form a join.A sequence σ = (v1, . . . , vn) is an h–extremal ordering of a graph G if for anyi = 1, . . . , n − 1 the vertex vi is h–extremal in Gi := G({vi, . . . , vn}) (inducedsubgraph of G formed by vertices of {vi, . . . , vn}). A graph G is homogeneouslyorderable if G has an h–extremal ordering.

In [4] it is proved that a graph G is homogeneously orderable if and only if thesquare G2 of G is chordal and each maximal two–set of G is join–split. This localstructure of homogeneously orderable graphs implies a simple O(n3) recognitionalgorithm of this class, which uses the chordality of the squares (see [4]). Sincethe square G2 of a graph G is chordal if and only if the graph L(D(G)) is chordal[3], we can reformulate that characterization in the following way.

Theorem 1. [4] A graph G is a homogeneously orderable graph if and only ifthe graph L(D(G)) of G is chordal and each maximal two-set of G is join-split.

This characterization will be very useful in Section 3 and Section 4. In Section 4,the following theorem from [9] will also be of use.


Theorem 2. [9] For any homogeneously orderable graph G = (V, E) with vertexfunction r : V → N and for any subset S of V , we have that S is r-dominatedby some clique C of G (i.e, dG(v, C) ≤ r(v) for every v ∈ S) if and only ifdG(x, y) ≤ r(x) + r(y) + 1 for all x, y ∈ S.

Finally, it is well known that the following lemma for chordal graphs holds.

Lemma 1 (Cycle Lemma for Chordal Graphs). Let C = (v0, . . . , vi−1, vi,vi+1, . . . , vk−1) be a cycle of a chordal graph G with k ≥ 4. Then, for any vertexvi of C, vivj ∈ E(G), for j = i − 1, i, i + 1 (mod k), or vi−1vi+1 ∈ E(G).

3 Additive Tree 3-Spanners

In this section, we show that every homogeneously orderable graph G admitsan additive tree 3-spanner. We prove that an algorithm similar to one presentedin [2] for constructing an additive tree 3-spanner of a dually chordal graph con-structs an additive tree 3-spanner of a homogeneously orderable graph. However,its correctness proof (Lemma 3 and Lemma 4) for homogeneously orderablegraphs is more involved.

Assume u is an arbitrary vertex of a homogeneously orderable graph G and k ∈{1, 2, . . . , ecc(u)}, where ecc(u) = max{dG(u, v) : v ∈ V (G)}. Let F k

1 , . . . , F kpk

be the connected components of the graph obtained from G by removing verticesof Dk−1(u). Denote Sk

i = F ki ∩ Nk(u) and Mk−1

i = N(Ski ) ∩ Nk−1(u). Mk−1

i iscalled the projection of Sk

i to layer Nk−1(u). Clearly, any two vertices x, y ∈ Ski

are connected by a path outside the disk Dk−1(u). Denote by Hk−1 the graphwith

⋃pk

i=1 Mk−1i as the vertex set and two vertices x, y are adjacent in Hk−1 if

and only if they belong to a common set Mk−1i .

Lemma 2. Every connected component A of the graph Hk is a two-set in G.

Proof. Let x, y be two vertices of a connected component A of the graph Hk.Then, we can find a collection of projections Mk

i1 , Mki2 , . . . , M

kih

such that x ∈Mk

i1, y ∈ Mk

ihand Mk

ij∩Mk

ij+1 = ∅ for all j = 1, . . . , h−1. Pick zj ∈ Mk

ij∩Mk

ij+1,

j = 1, . . . , h − 1 and let z0 := x and zh := y. Since zj−1, zj ∈ Mkij

, j = 1, . . . , h,

we can find two vertices v′j , v′′j ∈ Sk+1

ijadjacent to zj−1 and zj , respectively.

Let Pj be a path of F k+1ij

connecting the vertices v′j and v′′j . The disk Dk−1(u)together with D1(x), D1(y) and the disks of the family {D1(z) : z ∈ ∪h

j=1Pj}forms a cycle in the intersection graph L(D(G)). From chordality of this graph(see Theorem 1) and since Dk−1(u) ∩ D1(z) = ∅ holds for all z ∈ ∪h

j=1Pj , wededuce (see the Cycle Lemma for Chordal Graphs) that D1(x) ∩ D1(y) = ∅, i.e.dG(x, y) ≤ 2. ��

Next, we are going to show that for every connected component A of the graphHk there is a vertex z ∈ Nk−1(u) such that A ⊆ N(z). To show that, thefollowing lemma is needed, proof of which is omitted in this version.


Lemma 3. If k ≥ 2 and A is a connected component of the graph Hk, thenthere is a vertex t ∈ Nk−2(u) such that A ⊆ N2(t).

Now, we are ready to prove the following lemma.

Lemma 4. For every connected component A of the graph Hk there is a vertexz ∈ Nk−1(u) such that A ⊆ N(z).

Proof. If k = 1, then the lemma clearly holds. Hence, assume k ≥ 2. Accordingto Lemma 3, there is a vertex t ∈ Nk−2(u) such that A ⊆ N2(t). Hence, A∪{t} isa two-set of G. Let U be a maximal two-set of G such that A ⊆ U . By Theorem1, U consists of two subsets U1 and U2 and U1 �� U2. Clearly, A ∪ {t} must bein either U1 or U2, since dG(x, t) = 2 for each x ∈ A. Without loss of generality,assume A ∪ {t} is in U1. Then, U2 must contain a vertex z which is adjacent toall the vertices in A∪ {t}. This vertex z can only be in Nk−1(u). This concludesour proof. ��

From the above lemmata, one can give the following linear time algorithm toconstruct an additive tree 3-spanner for a homogeneously orderable graph G.

PROCEDURE 1. Construct an additive tree 3-spanner for a HOG G

Input: A homogeneously orderable graph G = (V, E).Output: An additive tree 3-spanner T of G.Method:

set E′ = ∅;pick an arbitrary vertex u in G;set i = ecc(u);for each vertex x ∈ Ni(u) do

arbitrarily choose a vertex y ∈ Ni−1(u) such that xy ∈ E(G);add xy into E′;

set i to i − 1;while i ≥ 1 do

construct the graph Hi and find its connected components;for each connected component A of Hi do

find a vertex z ∈ Ni−1(u) such that A ⊆ N(z);for each vertex x ∈ A add xz into E′;

for other vertices x which are in Ni(u) but not in Hi;arbitrarily choose a vertex y ∈ Ni−1(u) such that xy ∈ E(G);add xy into E′;

output T = (V, E′).

It is not hard to show that Procedure 1 can be implemented to run in lineartime. The most complex step is the construction of the connected componentsof the graphs Hi (i = ecc(u) − 1, . . . , 2, 1). We start from the layer Nk(u),where k = ecc(u), find its connected components F k

1 , . . . , F kpk

and contract eachof them into a vertex. Then find the connected components F k−1

1 , . . . , F k−1pk−1

in the graph induced by Nk−1(u) and the set of contracted vertices. To findthe connected components of the graph Hk−1, we construct a special bipartite


graph Bk−1 = (W, K; U). In this graph W = {f1, . . . , fpk} (a vertex fj repre-

sents component F kj ), and K is the vertex set of Hk−1 (which is

⋃pk

i=1 Mk−1i =

⋃pk

i=1(N(F ki ) ∩ Nk−1(u)). A vertex fj ∈ W and a vertex v ∈ K are adjacent

in Bk−1 if and only if v ∈ N(F kj ). The graph Bk−1 can be constructed in

O(∑

v∈Nk(u)∪Nk−1(u) deg(v)) time. Note that a vertex v ∈ Nk−1(u) belongs toHk−1 if and only if it has a neighbor in Nk(u). The connected components ofHk−1 are exactly the intersections of the connected components of Bk−1 withthe set K. After performing for Hk−1 all operations prescribed in the other linesof Procedure 1, we contract each of F k−1

1 , . . . , F k−1pk−1

into a vertex and descendto the lower level. We repeat the above until we come to the vertex u.

Theorem 3. The spanning tree T constructed by Procedure 1 is an additive tree3-spanner of G.

Proof. Let x, y be two arbitrary vertices of G. Assume x ∈ Nlx(u) and y ∈Nly(u). Without loss of generality, assume lx ≤ ly. Let P = (x0 = x, x1, · · · , xp =y) be a shortest path between x and y in G. Let k be the smallest integer suchthat P ∩ Nk(u) = ∅. There are two cases to consider.Case 1: There is exactly one vertex xi in P ∩ Nk(u).First, consider the subcase when i = 0, i.e., xi = x0 = x. Let PT be the pathbetween y and u in T . Let y′ be the vertex of PT from Nk(u). Clearly, x and y′

belong to the projection Mkj of that connected component F k+1

j (of the inducedsubgraph of G formed by vertices V \Dk(u)) which contains vertex y. By the wayT was constructed, dT (x, y′) ≤ 2 and dT (y, y′) = ly − lx must hold. This impliesdT (x, y) ≤ dT (x, y′)+dT (y′, y) ≤ 2+ly−lx ≤ 2+dG(x, y) since dG(x, y) ≥ ly−lx.

Let now i = 0. Since xi is the only vertex in P ∩Nk(u), xi−1 and xi+1 must bein Nk+1(u). Let P ′

T be the path between x and u and PT be the path between yand u in T . Let x′ and y′ be the vertices of P ′

T and PT taken from Nk(u). Clearly,x′ and xi belong to the projection Mk

jxof that connected component F k+1

jx(of

the induced subgraph of G formed by vertices V \ Dk(u)) which contains vertexx, and y′ and xi belong to the projection Mk

jyof that connected component

F k+1jy

which contains vertex y (it is possible that F k+1jy

= F k+1jx

). Since theseprojections Mk

jyand Mk

jxshare a common vertex xi, they must belong to the

same connected component of the graph Hk. By the way T was constructed,dT (x′, y′) ≤ 2 and dT (y, y′) = ly − k, dT (x, x′) = lx − k must hold. This impliesdT (x, y) ≤ dT (x, x′) + dT (x′, y′) + dT (y′, y) ≤ lx − k + 2 + ly − k ≤ dG(x, y) + 2since dG(x, y) = dG(x, xi) + dG(xi, y) ≥ lx − k + ly − k.Case 2: There are at least two vertices xi, xj in P ∩ Nk(u).In this case, dG(x, y) > lx −k+ ly −k (i.e., dG(x, y) ≥ lx + ly −2k+1) must hold.Again, let P ′

T be the path between x and u and PT be the path between y and uin T . Let x′ and y′ be the vertices of P ′

T and PT , respectively, taken from Nk(u).Clearly, x′, y′, xi and xj are in one connected component F k

t of the inducedsubgraph of G formed by vertices V \ Dk−1(u). By the way T was constructed,this implies dT (x′, y′) ≤ 4. Hence, dT (x, y) ≤ dT (x, x′) + dT (y, y′) + dT (x′, y′) ≤


ly + lx − 2k + 4 ≤ dG(x, y) + 3. This proves the second case and concludes theproof of the theorem. ��

Note that in [2] an example of a dually chordal graph (and hence of a homo-geneously orderable graph) is presented which does not have any additive tree2-spanner. Thus, the additive stretch factor 3 for the tree spanners presented inTheorem 3 is best possible if one wants to achieve it only with one tree.

4 Collective Additive Tree 2-Spanners

In this section, we show that every homogeneously orderable graph H admits asystem of O(log n) collective additive tree 2-spanners. According to [4], if a graphH is homogeneously orderable, then G = H2 is a chordal graph. Unfortunately,the method developed in [12] for constructing collective tree spanners in somehereditary classes of graphs (so-called (α, r)–decomposable graphs) cannot bedirectly applied to the class of homogeneously orderable graphs as this class is nothereditary (see [4]). We will work first on the square G = H2 of a homogeneouslyorderable graph H and then will move down to the original graph H . To the bestof our knowledge this is the first non-trivial result on collective tree spanners ofa non-hereditary family of graphs.

The following theorem is known for chordal graphs.

Theorem 4. [15] Every n-vertex chordal graph G contains a maximal clique Ssuch that if the vertices in S are deleted from G, every connected component inthe graph induced by any remaining vertices is of size at most n/2.

A linear time algorithm for finding for a chordal graph G a separating cliqueS satisfying the condition of the theorem is also given in [15]. We will call S abalanced separator of G.

Using the above theorem, one can construct for any chordal graph G a (rooted)balanced decomposition tree BT (G) as follows. If G is a complete graph, thenBT (G) is a one-node tree. Otherwise, find a balanced separator S in G, whichexists according to Theorem 4. Let G1, G2, . . . , Gp be the connected componentsof the graph G \ S obtained from G by removing vertices of S. For each graphGi (i = 1, . . . , p), which is also a chordal graph, construct a balanced decompo-sition tree BT (Gi) recursively, and build BT (G) by taking S to be the root andconnecting the root of each tree BT (Gi) as a child of S. Clearly, the nodes ofBT (G) represent a partition of the vertex set V of G into clusters S1, S2, . . . , Sq

which are cliques. For a node X of BT (G), denote by G(↓X) the (connected)subgraph of G induced by vertices

⋃{Y : Y is a descendent of X in BT (G)}

(here we assume that X is a descendent of itself).Consider two arbitrary vertices x and y of a chordal graph G and let S(x)

and S(y) be the nodes of BT (G) containing x and y, respectively. Let alsoNCABT (G)(S(x), S(y)) = Xt be the nearest common ancestor and X0, X1, . . . ,Xt be the sequence of common ancestors of S(x) and S(y) in BT (G) startingfrom the root X0 of BT (G). We will need the following lemma from [12].


Lemma 5. [12] Any path PGx,y, connecting vertices x and y in G, contains a

vertex from X0 ∪ · · · ∪ Xt.

Let now H = (V, E) be a homogeneously orderable graph. By Theorem 1, G =H2 is a chordal graph. We construct a rooted balanced decomposition tree BT (G)for G as described above. It is also known that any maximal clique of G is ajoin-split two–set of H . For any maximal clique C of G, let C = U1 ∪ U2 suchthat, in H , U1 � U2. Note that Xi (i = 0, 1, . . . , t) may not be a maximal cliquein G. Let C′

i be a maximal clique of G such that Xi ⊆ C′i. Assume C′

i = U i1 ∪U i

2with U i

1 � U i2 in H .

There are two cases to consider:

(a) U i1 ∩ Xi = ∅ and U i

2 ∩ Xi = ∅;(b) either U i

1 ∩ Xi = ∅ or U i2 ∩ Xi = ∅ (without loss of generality, we assume

U i1 ∩ Xi = ∅ in this case).

For each connected graph G(↓Xi), let H(↓Xi) be the induced subgraph ofH which has the same vertex set as G(↓Xi). Note that H(↓Xi) may not beconnected as not all edges of G(↓Xi) are present in H(↓Xi). We construct a treeTi for H(↓Xi) in the following way.

If U i1 ∩ Xi = ∅ and U i

2 ∩ Xi = ∅ holds, then we choose two vertices r1 ∈Xi ∩U i

1, r2 ∈ Xi ∩U i2 and add r1r2 into E(Ti). For each x ∈ Xi ∩U i

1, we add xr2into E(Ti). For each y ∈ Xi ∩ U i

2, we add yr1 into E(Ti). Then we extend thetree constructed so far by building a breadth-first-search-tree in H(↓Xi) startingat set Xi and spanning that connected component of H(↓Xi) which contains r1and r2.

If U i1 ∩ Xi = ∅ holds, then we choose a vertex r from U i

1 (which may not beeven in H(↓Xi)) and construct a tree Ti as follows. For every vertex x ∈ Xi, weput xr into E(Ti). Then, we extend the tree constructed so far by building abreadth-first-search-tree in H(↓Xi) starting at set Xi and spanning the verticesthat can reach r via vertices in H(↓Xi).

Below we will prove that for some vertices x, y ∈ H(↓Xi) the tree Ti willguarantee dTi(x, y) ≤ dH(x, y) + 2. Let x ∈ V (H(↓Xi)) and x′ ∈ Xi be verticessuch that dTi(x, x′) = dH(↓Xi)(x, Xi). Vertex x′ is called the projection of x inTi. We use PTi(x) to denote the shortest path between x′ and x in Ti.

Consider any two vertices x, y ∈ V . Let as before X0, X1, . . . , Xt be the com-mon ancestors of S(x) and S(y) in BT (G), and let Xi be the common ancestorwith minimum i such that there is a shortest path PH

x,y between x and y in H

with PHx,y ∩Xi = ∅. It is easy to see that such Xi must exist. Indeed, according to

Lemma 5, any path PGx,y between x and y in G must intersect X0 ∪ · · · ∪ Xt and

any path PHx,y is also a path between x and y in G (as G = H2 and therefore the

edge set of H is a subset of the edge set of G). By the choice of Xi, any shortestpath in H connecting x and y must be in H(↓Xi). Let Ti be the tree constructedas above for H(↓Xi). The following lemma can be proved to be true.

Lemma 6. dTi(x, y) ≤ dH(x, y) + 2 must hold.


Proof. Let PHx,y be a shortest path of H connecting x and y. By the choice of

Xi, this path is entirely in H(↓Xi) and intersects Xi. Let zx, zy be vertices ofthis path and Xi closest to x and y, respectively.

Let x′, y′ be the projections of x and y in Ti. According to the way Ti was con-structed, dTi(x′, y′) ≤ 3, dTi(x, x′) = dH(↓Xi)(x, x′) ≤ dH(↓Xi)(x, zx) = dH(x, zx)and dTi(y, y′) = dH(↓Xi)(y, y′) ≤ dH(↓Xi)(y, zy) = dH(y, zy). If dTi(x

′, y′) ≤ 2 orzx = zy, then we are done since dTi(x, y) ≤ dTi(x, x′) + dTi(x′, y′) + dTi(y, y′) ≤dH(x, zx)+dTi (x′, y′)+dH(y, zy) = dH(x, y)−dH(zx, zy)+dTi (x′, y′) ≤ dH(x, y)+2. Hence, assume that dTi(x′, y′) = 3 and dH(x, y) = dH(x, zx) + dH(y, zy),i.e., zx = zy. Denote s := zx = zy, dH(x, s) := lx and dH(y, s) := ly. SincedTi(x′, y′) = 3, by the way Ti was constructed, we have U i

1 ∩ Xi = ∅ andU i

2 ∩ Xi = ∅. Furthermore, vertices x′ and y′ can not be both in U i1 ∩ Xi or

both in U i2 ∩ Xi (otherwise, dTi(x′, y′) ≤ 2). Without loss of generality, assume

x′ ∈ U i1 ∩ Xi, y′ ∈ U i

2 ∩ Xi and s ∈ U i1 ∩ Xi. Consider the following radius func-

tion for vertices s, y, y′ in H : r(s) = 0, r(y′) = 0 and r(y) = ly − 1. According toTheorem 2, there is a clique C in H such that C r-dominates the set {s, y′, y}.Hence, there must exist a vertex y′′ ∈ (C \ Xi) such that sy′′, y′y′′ ∈ E(H) anddH(y′′, y) = ly − 2. This implies that y′′ is on a shortest path between x andy in H and, by the choice of Xi, y′′ is in H(↓Xi). Since sy′′, y′y′′ ∈ E(H), wehave dH(y′′, z) ≤ 2 for any vertex z ∈ U i

1 ∪ U i2. Therefore, {y′′} ∪ Xi is a clique

in G(↓Xi). The latter contradicts with the fact that Xi was a maximal clique inG(↓Xi) (see Theorem 4 and paragraph on the construction of BT (G) after thattheorem). Thus, the case that dTi(x′, y′) = 3 and zx = zy is impossible. Thisconcludes our proof. ��

Let now Bi1, . . . , B

ipi

be the nodes on depth i of the tree BT (G). For each sub-graph Hi

j = H(↓Bij) of H (i = 0, 1, . . . , depth(BT (G)), j = 1, 2, . . . , pi), denote

by T ij the tree constructed for Hi

j as above. Clearly, T ij can be constructed in

linear time. We call T ij a local tree of H . Since any two local trees T i

j and T ij′

share at most one vertex, T i =⋃

{T ij , j = 0, . . . , pi} is a forest. T i can be ex-

tended to T ′i to span all the vertices in H . Tree T ′i is called a global spanningtree of H . By Lemma 6, we immediately get the following result.

Lemma 7. For any two vertices x, y ∈ V , there exists a global spanning tree T ′i

such that dT ′i (x, y) ≤ dH(x, y) + 2.

By the way BT (G) was constructed, the depth of BT (G) is at most log2 n.Hence, there are at most log2 n global spanning trees of H . Obviously, G = H2

can be obtained in O(nm) time. According to [12], BT (G) can be constructed inO(n2 log2 n) time (note that G may have O(n2) edges). Each T ′i is constructiblein linear time. Therefore, the following theorem is true.

Theorem 5. Any n-vertex homogeneously orderable graph admits a system oflog2 n collective additive tree 2-spanners which can be constructed in O(nm +n2 log2 n) time.


5 Conclusion

In this paper we studied collective additive tree spanners in homogeneously or-derable graphs. We showed that every n-vertex homogeneously orderable graphadmits an additive tree 3-spanner constructible in linear time and a system ofO(log n) collective additive tree 2-spanners constructible in polynomial time.These results generalize known results on tree spanners of dually chordal graphsand of distance-hereditary graphs. We mentioned also that there are homo-geneously orderable graphs which do not admit any additive tree 2-spanners.Hence, it is natural to ask how many spanning trees are necessary to achievecollective additive stretch factor of 2 in homogeneously orderable graphs. Weknow that this number is not a constant (a proof of this is presented in the fullversion of the paper) and is not larger than log2 n. One may ask also how manyspanning trees are necessary and how many are sufficient to achieve collectiveadditive stretch factor of 1 or 0 in homogeneously orderable graphs. The answerto this question is simple. Any graph G on n vertices admits a system of atmost n − 1 collective additive tree 0-spanners. On the other hand, it is easyto see that any system of collective additive tree 1–spanners of homogeneouslyorderable graphs will need to have at least Ω(n) spanning trees (a proof of thisis presented in the full version).

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